Kyriakos Kalorkoti

Rewrite Systems, monoids and free resolutions

A monoid is simply a set of strings over an alphabet with a set of
equalities (from which other equalities follow by the obvious notion of equivalence). It is a very old result that the word problem for
monoids has no algorithmic solution in general. However if we have a
finite convergent rewrite system for a monoid then it has solvable word
problem and (assuming a finite alphabet) a finite presentation. For a
long time the question of whether the converse is true was open (i.e.,
if a monoid has a finite presentation and solvable word problem does it
have a finite convergent rewrite system?). In the late 1980's and
early 1990's it was realized that homological methods could be used to
resolve the question. In this talk I will give an overview of the
problem and the methods used, keeping to general description rather
than technical details.